The Parametric Complexity of Operator Learning

TL;DR

This paper investigates the limitations of neural operator architectures in approximating general operators, revealing a curse of parametric complexity, but also introduces a specialized neural operator, HJ-Net, that overcomes this challenge for Hamilton-Jacobi equations.

Contribution

The paper proves a curse of parametric complexity for general operators and introduces HJ-Net, a neural architecture that overcomes this curse for Hamilton-Jacobi solution operators.

Findings

General operator learning suffers from an infinite-dimensional curse of complexity.

HJ-Net can provably outperform existing neural operators on Hamilton-Jacobi problems.

Error and complexity bounds demonstrate HJ-Net's effectiveness in overcoming the curse.

Abstract

Neural operator architectures employ neural networks to approximate operators mapping between Banach spaces of functions; they may be used to accelerate model evaluations via emulation, or to discover models from data. Consequently, the methodology has received increasing attention over recent years, giving rise to the rapidly growing field of operator learning. The first contribution of this paper is to prove that for general classes of operators which are characterized only by their $C^r$- or Lipschitz-regularity, operator learning suffers from a "curse of parametric complexity", which is an infinite-dimensional analogue of the well-known curse of dimensionality encountered in high-dimensional approximation problems. The result is applicable to a wide variety of existing neural operators, including PCA-Net, DeepONet and the FNO.The second contribution of the paper is to prove that this general curse can be overcome for solution operators defined by the Hamilton-Jacobi equation; this is achieved by leveraging additional structure in the underlying solution operator, going beyond regularity. To this end, a novel neural operator architecture is introduced, termed HJ-Net, which explicitly takes into account characteristic information of the underlying Hamiltonian system. Error and complexity estimates are derived for HJ-Net which show that this architecture can provably beat the curse of parametric complexity related to the infinite-dimensional input and output function spaces.